Any deep results from Hopf Algebras (or Quantum groups)?

[tim_lou]

Hi. I'm currently working on a expository paper about quantum groups and Hopf algebras. However, from all the books I've read, they are more about examples (deformations of various groups) than actual interesting results (Or perhaps I just don't understand them enough to draw any interesting results). I mean yes I know universal enveloping algebra of any lie algebra is a Hopf algebra naturally, but why would it pay to study them this way? Those books mentioned something about Yang Baxter equation but I haven't reached that level in physics yet.

I am having great difficulties trying to convince myself why quantum groups are interesting.
It would help me tremendously if someone can point out any deep theorems that can be proved in the frameworks of Hopf algebras (perhaps something to do with cohomology of lie algebras?). Or maybe physical applications of matrix quantum groups (instead of just pure mathematical jargon).

By deep theorems i mean things like embedding of manifolds in R^{2n+1}, Riez Representation theorem...things that are highly unobvious and/or enlightening.

Thanks for the help.

 

[fresh_42]

I know from own experiences that this is the wrong way to look at it. You have a description looking for what it describes! I don't think that universal enveloping algebras are of great use for otherwise detailed problems. Furthermore are Hopf algebras very mathematical constructions.

Wikipedia quotes Kassel's Quantum Groups
https://www.amazon.com/dp/1461269008/?tag=pfamazon01-20
which I think is the most promising approach.

Or a lecture note which goes in the direction of quantum computing:
https://www.math.uni-hamburg.de/home/schweigert/ws12/hskript.pdf